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Anomalous quantized conductance in a half-metal/topological superconductor/half-metal junction

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 175702 (http://iopscience.iop.org/0953-8984/26/17/175702) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 175702 (7pp)

doi:10.1088/0953-8984/26/17/175702

Anomalous quantized conductance in a half-metal/topological superconductor/ half-metal junction C D Ren1,K S Chan2 and and J Wang1 1

  Department of Physics, Southeast University, Nanjing 210096, People's Republic of China   Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, ­Kowloon, Hong Kong, People's Republic of China 2

E-mail: [email protected] Received 24 December 2013, revised 24 February 2014 Accepted for publication 3 March 2014 Published 11 April 2014 Abstract

The composite topological superconductor (TS), which is made of one-dimensional spin–orbit coupled nanowire with proximity-induced superconductivity from an s-wave superconductor, is not a pure p-wave superconductor, but has a suppressed s-wave pairing. We calculate the conductance spectrum of a half-metal/TS/half-metal junction in order to probe the pairing states and the spin texture of the p-wave pairing. It is found that, besides the regular quantized conductance peak contributed by Majorana fermions (MFs) when the half-metal magnetization is parallel to the MF spin, an anomalous quantized conductance peak exists when they are almost antiparallel. The physical origin is the MF-assisted local Andreev reflection to condense s-wave pairings. The anomalous quantized conductance is also confirmed by the Kitaev's p-wave model with a nonzero s-wave pairing. The findings might provide a new way to find the MF. Keywords: topological superconductor, conductance, Fano effect (Some figures may appear in colour only in the online journal)

1. Introduction

an s-wave pairing component but the effective p-wave one is dominative. Similar to a noncentrosymmetric superconductor, both the intraband p-wave and interband s-wave paring states exist and the properties of the system are determined by the dominative one. Actually, in the realistic case the pairing strength Δ is not negligible compared to the magnetic field as employed in recent experiments [8–10], so the s-wave component is sizable and will influence the MF transport properties in some way. With regard to detection of the MFs in the TS, one of the most direct ways is to perform a local tunneling measurement in a spinless metal/TS, which yields a quantized zerobias conductance peak. Recent tunneling experiments [8–10] showed positive results but the story is not over yet due to the controversy [11–18] about the origin of the zero-bias conductance peak. Note that the majority of studies to date

Solid state Majorana fermions (MFs), exotic quantum quasiparticles of their own antiparticle, have received much attention due to their underlying physics and their potential applications for fault-tolerant quantum computation [1, 2]. The most experimentally promising route for realizing solid state Majorana modes among many theoretical proposals involves using Rashaba-type spin–orbit coupling in combination with a conventional s-wave superconductor to produce a one-dimensional effective spinless topological superconductor (TS) [3–5]. In this nanowire scheme, it is well known that when the superconductor gap is much smaller than the applied magnetic field Δ  ≪  h, the upper evanescent band can be ignored and the system can be mapped to an effective spinless superconductor [6, 7]; however, there exists still 0953-8984/14/175702+7$33.00

1

© 2014 IOP Publishing Ltd  Printed in the UK

C D Ren et al

J. Phys.: Condens. Matter 26 (2014) 175702

z

y nanowire

superconductor

x

barrier

ferromagnetic ferromagnetic

nanowire

SOC nanowire insulator

ferromagnetic

insulator

insulator

V Figure 1.  Sketch of the proposed HM/TS/HM setup. The TS is an SOC nanowire sandwiched between an s-wave superconductor and a ferromagnetic insulator (the magnetization mt is along the z direction). The left and right HM sections are the magnetized nanowire, where there is no SOC or superconductivity. They connect to the TS via tunnel barriers.

concerning tunneling properties of topological superconductors simply assume the tunneling system is perfectly spinless and are only concerned with the charge degree of freedom in the TS [19–25], while the influence of the s-wave pairing on conductance and the spin texture of MFs are considered less important. This may produce some properties characterizing the MFs in the TS. In this work, we study the differential conductance in a halfmetal (HM)/TS/HM junction, where the TS is a spin–orbit coupling (SOC) semiconductor nanowire in contact with an s-wave superconductor and a ferromagnetic insulator [26] as schematically shown in figure 1. It is assumed that in the HM sections there is no SOC or superconductivity. It is shown that in the topological phase, there is a quantized conductance contributed by the MFs when the HM magnetization mh is nearly parallel to mt in the TS. While in the vicinity of antiparallel configuration, the conductance exhibits a valley G = 0 as well as an anomalous quantized peak G = 2e2/h provided that the s-wave pairing component Δs is not negligibly small. The conductance valley comes from the rigorous antiparallel configuration between the MF spin and mh. The anomalous conductance peak is attributed to the local Andreev reflection of the s-wave pairing with the help of MF excitation. We also employ a toy Kitaev's model including a nonzero s-wave component to demonstrate both the conductance valley and anomalous quantization peak.

H = HTS + HHM + HT (1) HTS =

∑ ci, σ† ( tσ0 − μσ0 − m tσz ) ci,σ i, σ

1 − ∑ ( tici,†σ ci+1, σ + h.c.) (2) 2 i, σ −

i 2

∑ ( tsoci†σ σyci+1, σ ′ + h.c.) + ∑ ( Δci, ↑ci, ↓ + h.c.)

i, σσ ′

i

HHM = ∑ d i,†σ ( tσ0 − μσ0 − mHM · σ ) d i, σ , (3) i, σ

HT = ∑ tL ( dL,† σc1, σ + h.c.) + tR ( dR,† σcN, σ + h.c.) (4) σ

where ci,†σ ( ci, σ ) and dσ† ( dσ ) are the creation (annihilation) operators of an electron with spin σ in the TS and HM, respectively; t denotes the hopping integral and tso is the SOC strength. The magnetization mt in the TS is assumed fixed along the z-axis, while the mh in the HFs are assumed to be along an arbitrary direction (sin α cos β, sin α sin β, cos α), where α is the polarization angle and β is the azimuthal angle. It should be noted that the topological phase transition condition mh > μ2 + Δ2 is for the infinite nanowire, but does not strictly apply to the finite one due to nonzero Majorana modes [18]. tL(R) is the coupling strength between the left (right) HM and TS and can also stand for the barrier strength between them. σi=x, y, z is the Pauli matrix and σ0 is the unit matrix. The scattering matrix is employed to calculate the conductance and with the help of the Green's function, the scattering matrix is given as [22],

2.  Model and Formalism We consider the HM/TS/HM junction shown in figure 1, in which the left and right HM sections are coupled to the TS via tunneling barriers and the HM magnetization can be rotated in space in order to study the noncollinear effect. A similar setup was also proposed in reference [25], where the role of varying the magnetization was to change the splitting energy of the MFs due to the spinful TS within the topological insulatorsuperconductor structure in the model. We describe this setup by using a tight-binding model in one dimension for noninteracting electrons,

Sijλρ = − δ ijδλρ + i[ Γiλ ]1/2 · G r · [ Γ ρj ]1/2 , (5)

which describes the scattering amplitude of a ρ particle from lead j to a λ particle in lead i. i and j denote the HF leads. λ, ρ  ∈  {e, h} denote the electron and hole channels. −1 r −1 G r = [ ( gTS ) − Σ r ] is the retarded Green's function of the 2

C D Ren et al

J. Phys.: Condens. Matter 26 (2014) 175702

0.15

(a)

0.1

G(2e2/h) 1.0 0.8 0.6 0.4 0.2 0

0.05 0 0.15

(b)

0.1 0.05 0

-0.1

0.05

0

0.25

0

0.1 0.5

(c) G(2e2/h)

G(2e2/h)

0.5

-0.05

0.2

(d)

0.25

0

0.15

0.15

0

0.15

Figure 2.  Conductance contour GL versus energy E and chemical potential μ for the polarization angle α = 0 in (a) and π in (b). The

differential conductance GL as a function of E is presented in (c) and (d) for the two cases with the chemical potential μ taken from the dotted lines in (a) and (b). The unit of chemical potential is t.

model, which can be calculated by the recursive iteration method. Γiλ = i[( Σiλ )r − ( Σiλ )a ] is the line-width function and ( Σiλ )r (a) is the λ particle retarded (advanced) self-energy of lead i. In terms of the scattering matrix method, the differential conductance of the lead i can be expressed as [27],



e 2 eV PijdE Sij = (8) h 0

with the definition ee ee ee eh he he eh hh hh hh P  ij( E ) = δ ijRii + δ ijRii − Rij Rji − Rij Rji + Rij Rji + Rij Rji . (9)

G i = I − Riiee − Rijee + Riihh + Rijhh , (6)

3.  Calculations and Discussions

where Rμνλρ ( E ) = Siλμe ( E )[ Siρνe ( E )]† (λ, ρ  ∈  {e, h}); Riiee and hh Rii represent the local normal reflection and local Andreev reflection, and Rijee and Rijhh stand for the nonlocal elastic cotunneling and crossed Andreev reflection of electrons from lead i to lead j, respectively. The conductance formula above provides full information about the physical processes with Majorana modes, in particular, reflecting their nonlocal nature. However, because the probabilities of transferring an electron or a hole from lead i to lead j and from lead j to lead i are the same due to the detailed balance, the crossed Andreev reflection cannot be detected directly by the conductance measurement. The short noise probe, however, can provide information on the correlations of the end Majorana modes and the charge of the individual carriers, which are not available in the usual conductance experiments. The short noise evaluation has already been proposed [28–30] to detect the tunnel processes with MFs. So it will also be calculated and used to analyze our results. The Fano factor (the ratio between shot noise to current) for each tunneling process is given by [27],

We focus on the differential conductance GL of the left lead as a function of the magnetization direction (α, β) of the HM sections. In the numerical calculations, we use a set of parameters consistent with the InSb properties as in reference [8], and choose the hopping energy as the energy unit t = 1, the induced pairing potential is Δ = 0.04t and the SOC is tso = 0.1t. The magnetizations in the TS and HM are also set to be equal, mt = mh = 0.08t, so that mt is strong enough to entirely project away the upper evanescent band and mh suffices the definition of a HM, i.e., mt(h) > |μ|, so the TS is in the topological nontrivial phase. The length of the TS wire is chosen as l = 40a with the lattice constant a. Due to the finite-length wire, the coupling between the two MFs causes a splitting of the zeroenergy mode in an infinite-length TS and the energy splitting (EM) is characterized by the oscillatory behavior depending on mt and μ [31]. In figure 2, the differential conductance GL is presented as a function of the chemical potential μ and electron incidence energy E at two different magnetization configurations: α = 0 and α  =  π. It is shown that the differential conductances at these two configurations have the same oscillating behavior, which is consistent with the energy splitting of MFs in the finite-length TS, and reflects the fact that both of them are directly associated with the physics of MFs. The resonant

Sij ( ω = 0) , Fij = e (7) ( I + Ij ) 2 i

where Sij(ω = 0) is the short noise given by 3

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J. Phys.: Condens. Matter 26 (2014) 175702

G 2e2 h

1

0.5 0 0

0

2 0.5

(b)

0.75

0.5

0

0.1 0

0

(d)

0.2

0

0

2

(e)

0.2 0.1 0

2

(c)

0.25

2

2 Rhh LR 2e h

2 Ree LR 2e h

2

2 Rhh LL 2e h

ee 2e2 h RLL

1

(a)

0

2

Figure 3.  Differential conductance GL (a) and scattering coefficients (b)–(e) as functions of the polarization angle α with fixed azimuthal

angle β = 0 (blue solid lines) and π/2 (red dashed lines). The chemical potential μ is chosen to be the same as that in figures 2(c) and (d) and the incident energy is equal to the splitting energy of MFs, E = EM, corresponding to the incident energy of the peaks in figures 2(c) and (d).

curve in figure 2(a) exhibits the quantized conductance e2/h contributed by MFs. Due to the finite-length TS, the local Andreev reflection and crossed Andreev reflection both contribute to the conductance. However, the crossed Andreev reflection is canceled by co-tunneling, so the nonzero-bias differential conductance is quantized to e2/h instead of 2e2/h. The results in figure 2(b) are in stark contrast to the situation of Δ ≪ mt, where the evanescent band Ψ _ + of the SOC nanowire is projected away due to μ residing in the gap opened by the Zeeman splitting mt, while the lower band Ψ _ − is nearly polarized along the direction of mt, i.e., Ψ _ − ≈ Ψ↓. As a result, the differential conductance GL in the antiparallel configuration of mh and mt disappears. In contrast, the nonzero resonant conductance approaching the quantized value in figure 2(b) is present in the antiparallel configuration, which is also related to MFs and indicates there are neglected factors contributing to this antiparallel conductance. Certainly, the difference of GL in figures 2(a) and (b) is clear: the halfheight width of the antiparallel configuration is much smaller than the parallel one, and the peak value of the antiparallel one is also less than the parallel one as seen in figure 2(c) and (d). It is known that the differential conductance is proportional to the local state density, G (eV)∝ N(E = eV), so the spin direction of MFs should not be exactly along the z-direction and the differential conductance should change with the direction of mh.

Next, we study the conductance GL as a function of polar angle α for two different fixed azimuthal angles, β = 0 and π/2, as shown in figure 3(a), where the fixed chemical potential μ and energy E are taken from the resonances at the vertical dashed line in figures 2(a) and (b). It can be seen that for β = π/2, the conductance is fully symmetric with a minimum at α = π. For β = 0, the conductance is nearly asymmetric with respect to α = π with a minimum GL = 0 at α ≃ 0.94π and a maximum GL(E) = 2e2/h at α ≃ 1.06π . The conductance valley comes from the fact that mh is exactly antiparallel to the MF spin since all the scattering coefee ficients except the normal reflection RLL vanish, as seen in figures 3(b)–(e). Therefore, the MF spin deviates slightly from α = 0. The quantized conductance peak at 1.06π was not expected since mt is almost antiparallel to mh. In order to verify this, we can check the four scattering coefficients in figure 3(b)–(e). For β = π/2, it is shown that all the four contributing terms are also symmetric with respect to α = π, and at this symmetrical ee point the normal reflection coefficient RLL does not reach a maximum and other processes are not completely suppressed. These indicate that the spin direction of MFs does not lie in the yz plane. For β  =  0, we can see that at α ≃ 0.94π, the incoming electrons are totally reflected by normal reflection and other processes vanish, which suggest that the spin direction of MFs should be exactly antiparallel to the magnetization direction of the HM, so the spin direction of MFs lies in the xz plane. 4

C D Ren et al

J. Phys.: Condens. Matter 26 (2014) 175702

(a)

1

s =0.02t

G(2e2 /h)

G(2e2 /h)

1

0.5

0

s =0

0.5

0

2

0

(b)

2

0

Figure 4.  Differential conductance GL versus the polarization angle α for the toy Kitaev's p-wave model. Parameters are β = 0 (blue

solid lines) and π/2 (red dashed lines), Δp,↓ = 0.04t, Δp,↑ = 0 and Δs = 0.02t in (a) and Δs = 0 in (b). The same strategy is used to select the chemical potential μ and the incident energy E = EM as in figure 3.

1

0.5

G(2e2 /h)

G(2e2 /h)

1

s =0.01t s =0.02t

0

s =0.03t

0

2

tso=0.15t 0

2

Figure 6.  Differential conductance GL versus the polarization angle

α. β = 0 and the other parameters are the same as those in figure 3.

polarization angle α for the Kitaev's model. β = 0 and the other parameters are the same as those in figure 4.

Zeeman splitting and μ residing in the gap opened by mt, Δp,+ is the topological trivial strong-pairing state and Δp,− is the topological nontrivial weak-pairing state. As discussed above, when mt ≫ Δ the upper evanescent band Ψ _ + can be safely projected away and only Δp,− is present in the TS. However, when Δ is the same order of magnitude as mt, the s-wave pairing component is sizable, so it may play an important role in forming the anomalous conductance peak at α ≃ 1.06π, as shown in figure 3(a). To further understand this naive guess, we employ Kitaev's p-wave model by including both the s-wave and strong p-wave pairing to replace the composite TS Hamiltonian of equation (2), which reads

We proceed with the discussion of the results for β = 0. At α ≃ 1.06π, the normal reflection coefficient reaches a minimum ee ( RLL = 0), as shown in figure 3(b), and the local Andreev reflection hh reaches a maximum RLL = 0.5(2e 2 / h ), as shown in figure 3(c), since the evanescent spin branch has always been normally reflected in the HM sections and only the conductive spin branch contributes to the local Andreev reflection. Besides, the two nonee hh local processes vanish, RLR = RLR = 0, as shown in figures 3(d) and (e). Therefore, the quantized conductance merely originates from the local Andreev reflection, which means that this anomalous conductance peak at α ≃ 1.06π should not stem uniquely from MFs. In order to account for the unexpected anomalous conductance peak, we first transform the Hamiltonian in equation (2) without a pairing potential into k-space as

(

tso=0.05t tso=0.10t

0

Figure 5.  Differential conductance GL as a function of the

HTS = ∑ ci,†σ ( tσ0 − μσ0 − m tσz ) ci,σ − i, σ

+ ∑ ( Δs ci, ↑ci, ↓ + h.c.) +

)

H0 = ∑ ϵ ( k ) + ν m t2 + ( αsock )2 ak,† νak, ν , (10) k , ν =±

0.5

1 ∑ ( tici,†σ ci+1, σ + h.c.) 2 i, σ

1 ∑ ( Δp, σ ci, σ ci+1, σ + h.c.), 2 i, σ

i (13)  where the Zeeman term mt breaks the spin degeneracy and renders one spin band active and the opposite spin band evanescent. Δs is the s-wave pair potential and Δp,σ are the counterparts of the p-wave pairing. With this toy model, all the results in figure 3 can be recovered and the anomalous conductance peak at α ≃ 1.06π appears with suitable parameters. Moreover, we found relations between the conductance and the three pairing states using this toy model: (1) without the evanescent band or strong pairing state Δp,↑, the anomalous conductance peak also appears as shown in figure 4(a); (2) without the nontrivial term Δp,↓, the differential conductance vanishes at arbitrary angles (not shown); (3) without the s-wave pair state, the anomalous conductance peak

and then the pairing potential Δ in equation (2) is transformed into three parts,

⎛ ⎞ m tΔ Δs = ∑ ⎜⎜ ak, +a−k, − + h.c. ⎟⎟ , (11) m t2 + ( αsock )2 ⎠ k ⎝ ⎛ ⎞ − iναsockΔ Δp, ν = ∑ ⎜⎜ ak, νa−k, ν + h.c. ⎟⎟ , 2 2 (12) ⎠ k , ν =± ⎝ 2 m t + ( αsock )

where Δs is the interband s-wave pairing and Δp,ν=± are the p-wave interband parings with opposite chirality. Due to the 5

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J. Phys.: Condens. Matter 26 (2014) 175702

Fano factor

2

F LR

1

(b)

1

0

0 0

2 Fano factor

2

F LL

(a)

0.1

0.2

0 2

(c)

1

1

0

0 0

0.1

0.1

0.2

0.2

(d)

0

0.1

0.2

Figure 7.  Fano factors FLL (blue solid lines) and FLR (red dashed lines) as a function of incident energy E for different polarization angle α. β = 0 and other parameters are the same as those for figure 3.

the MF-assisted crossed Andreev reflection, whereas the local Andreev reflection is completely suppressed. As E increases from 0 to the splitting energy EM of the two end MFs, the increase of FLL and decrease of FLR are due to the growth of the local Andreev reflection with E. For the α ≈ 0.94π case shown in figure 7(b), the crossed Fano factor FLR has the opposite behavior compared with the α = 0 case for E 

half-metal junction.

The composite topological superconductor (TS), which is made of one-dimensional spin-orbit coupled nanowire with proximity-induced superconductivity f...
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